Q39. - Algebra 2

Question

For Exercises $39-41$, use the following information.

A marker buoy off the coast of Gulfport, Mississippi, bobs up and down with the

waves. The distance between the highest and lowest point is $4$ feet. The buoy moves

from its highest point to its lowest point and back to its highest point every $10$ seconds.

Write an equation for the motion of the buoy. Assume that it is at equilibrium at

$t=0$ and that it is on the way up from the normal water level.

Step-by-Step Solution

Answer

The function is $y=2\sin \frac{\pi }{5}t$

Step 1. Given information

A bouy with difference of $4$ feet is given

Step 2. Concept used

The motion is periodic in nature. By using trigonometric functions we can write the equation of the motion as described in the question.

Step 3. Calculation

They had given at $t=0$ the height is zero.

So, we use sine function-

the general equation can be written as-

$y=A\sin \omega t$

Amplitude can be written as-

 $\begin{array}{l}a=\frac{4}{2}\\ a=2\end{array}$

The period is ten seconds, therefore the period is-

 $\begin{array}{l}t=10\\ \omega =\frac{2\pi }{t}\\ \omega =\frac{2\pi }{10}\\ \omega =\frac{\pi }{5}\end{array}$

The equation is-

 $y=2\sin \frac{\pi }{5}t$